In the following question, pick out the most suitable substitute for the question mark: 6 : 222 :: 25 : ?

Introduction / Context:

This is an analogy based numerical reasoning question. The relation between the first pair of numbers, 6 and 222, must be replicated for the second pair, 25 and the unknown value. Such questions are common in reasoning sections where the candidate must detect a mathematical relationship and then apply it consistently to a new number.

Given Data / Assumptions:

• The analogy is written as 6 : 222 :: 25 : ?.
• The same mathematical operation that links 6 to 222 must link 25 to the required answer.
• The operation may involve powers, addition, or other standard arithmetic operations.
• Only one answer option will respect the exact relationship.

Concept / Approach:

A good way to approach such analogies is to try simple operations on the first number and see if they match the second. Since 222 is much larger than 6, exponentiation is a natural guess. We can test whether 222 can be expressed in terms of 6^2, 6^3, or similar, and then see what analogous operation on 25 produces one of the answer choices. In this case, the combination of cube and addition works elegantly and yields a neat rule.

Step-by-Step Solution:

Step 1: Compute 6^3 = 6 * 6 * 6 = 216. Step 2: Add 6 to this cube: 216 + 6 = 222. So 6 is mapped to 222 using the operation n^3 + n. Step 3: Apply the same operation to 25: compute 25^3. Step 4: 25^3 = 25 * 25 * 25 = 15625. Step 5: Add 25 to this result: 15625 + 25 = 15650. This must be the number that stands in the same relationship to 25 as 222 does to 6.

Verification / Alternative check:

We can express the mapping symbolically as f(n) = n^3 + n. Then f(6) = 6^3 + 6 = 216 + 6 = 222, which matches the given pair. For the second part, f(25) = 25^3 + 25 = 15625 + 25 = 15650. This exactly matches one of the answer options and preserves the same functional relationship. There is no simpler operation of similar elegance that maps 6 to 222 and also leads to one of the provided options when applied to 25.

Why Other Options Are Wrong:

Option 650 is far too small compared to 25^3 and does not arise from the operation n^3 + n. Option 15625 equals 25^3, but that would correspond to the rule n^3, which fails for the first pair since 6^3 is only 216, not 222. Option 15600 cannot be written as 25^3 plus or minus 25 in a neat way that keeps the relationship symmetrical. Option 3750 is also not consistent with the rule that 6 behaves as 6^3 plus itself. Therefore, only 15650 maintains the exact pattern n^3 + n.

Common Pitfalls:

Candidates sometimes focus on addition or multiplication alone and miss exponentiation, especially when the second number is significantly larger. Another mistake is to force unrelated operations for the two pairs instead of seeking one unifying rule. Some may stop after seeing that 15625 matches 25^3 and ignore the fact that 6^3 would then not match 222. Always confirm that the same function applied to both numbers in the first pair works for the second pair as well.

Final Answer:

The value that stands to 25 as 222 stands to 6 under the rule n^3 + n is 15650.